# Null

0th

Percentile

##### Null Spaces of Matrices

Given a matrix, M, find a matrix N giving a basis for the (left) null space. That is crossprod(N, M) = t(N) %*% M is an all-zero matrix and N has the maximum number of linearly independent columns.

Keywords
algebra
##### Usage
Null(M)
##### Arguments
M
Input matrix. A vector is coerced to a 1-column matrix.
##### Details

For a basis for the (right) null space ${x : Mx = 0}$, use Null(t(M)).

##### Value

• The matrix N with the basis for the (left) null space, or a matrix with zero columns if the matrix M is square and of maximal rank.

##### References

Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth edition. Springer.

qr, qr.Q.

• Null
##### Examples
# The function is currently defined as
function(M)
{
tmp <- qr(M)
set <- if(tmp$rank == 0L) seq_len(ncol(M)) else -seq_len(tmp$rank)
qr.Q(tmp, complete = TRUE)[, set, drop = FALSE]
}
Documentation reproduced from package MASS, version 7.3-40, License: GPL-2 | GPL-3

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